The actin filamant system of eukaryotic cells is the dynamical structure primarily responsible for the so called "amoeboid" motions: phagocytosis, cytokinesis, pseudopodal extension/retraction, cytoplasmic streaming, cell spreading and crawling locomotion on surfaces. The advent of supercomputers and the associated technology for numerically solving systems of nonlinear partial differential equations makes it possible to investigate the properties of realistic continuum mechanical models of the actin filament system. This means that, for the first time, the detailed comparison of theories of cell motion with the results of observation and experiment can be carried through. The potential impact of continuum mechanics and supercomputers on biological understanding of the cytoskeleton and amoeboid motility can be gauged by reference to the revolutionary changes currently underway in the area of macro-molecular structure and function. To realize these potentialities, however, existing theoretical work must be greatly extended. In the past, computational methods for solving models of the cytoplasmic mechanics have been restricted to the confined motions of isotropic contractile networks (i.e., motions within a fixed spatial domain in which the actin filaments are randomly orientated). As a results, these methods could not be applied to many of the most interesting forms of cell motility (e.g., pseudopodal extension). It is now proposed to improve existing continuum models and associated numerical techniques so as to remove these limitations. We propose to demonstrate the practical utility of our theoretical and computational methodologies by carrying out detailed mechanical analyses of experimental systems for which good data are available. The first such analysis will be of the problem of the stability and self assembly of the monopodial form and the connected problem of pseudopodial extension in the giant free living amoebae such as A. proteus. Subsequently, we propose to analyze the dynamics of the leading lamella of tissue culture cells such as fibroblasts.